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This chapter begins with an introduction to multivariate procedures, which allow social workers and other human services researchers to analyze multidimensional social problems and interventions in ways that minimize oversimplification. Examples of multivariate statistical procedures to predict and describe relationships include multivariate multiple regression (MMR), multivariate analysis of variance (MANOVA), and multivariate analysis of covariance (MANCOVA). Structural equation modeling (SEM) may be used for data simplification and reduction, description, and prediction. The discussion then turns to the rationale for multivariate analysis followed by a description of the organization and contents of this book.

This chapter aims to review how Bayesian inference can be applied to some of the so-called systems of equations models. These models can be defined in several forms including multivariate regression models, vector autoregressive (VAR) models, simultaneous equation models (SEM), and systems of seemingly unrelated regression equation (SURE) models. This chapter analyses VAR models which are formally equivalent to multivariate regression models and suggests that VAR models can be either open or closed depending on whether exogenous variables are included or not.

This chapter presents the modeling results and their interpretation. First, the synthesis of success factors from existing success factor syntheses is developed and theoretically motivated. Then, the descriptive statistics and correlations between success factors are described analogously for each data set (CPR, NIIS, IFRI, and an overall model from all data sets). Finally, for each modelling methodology (multivariate regressions, random forests, and neural network), the model qualities are presented. In addition, the individual factors are described according to their importance for ecological success. Each presentation of results is followed by a discussion. The chapter is concluded with robustness and sensitivity analyses.

This chapter summarizes similarities and differences between multivariate analysis of variance (MANOVA), multivariate analysis of covariance (MANCOVA), multivariate multiple regression (MMR), and structural equation modeling (SEM). It offers suggestions to guide their differential use. It also compares and contrasts MANOVA and MANCOVA versus MMR, MANOVA and MANCOVA versus SEM, and MMR versus SEM.

This chapter begins with a set of four lemmas from elementary multivariate regression which provides the essentials of the theory for the general linear state space model from both a classical and a Bayesian standpoint. The four lemmas lead to derivations of the Kalman filter and smoothing recursions for the estimation of the state vector and its conditional variance matrix given the data. The chapter also derives recursions for estimating the observation and state disturbances, and derives the simulation smoother, which is an important tool in the simulation methods employed later in the book. It shows that allowance for missing observations and forecasting are easily dealt with in the state space framework.

The three statistical analysis methods used are presented: multivariate linear regression, random forests, and artificial neural networks. Their respective advantages and disadvantages are discussed as well as how they can complement each other. Using three independent methods increases the robustness of results considerably. A further subchapter describes the operationalization of success factors through the development of an indicator system. Particular attention is paid to the validation of this system through external experts, its practical use, and operationalization of ecological success. Different ways of operationalizing ecological success are compared for forests. The surprising conclusion is that experts’ judgment is equivalent to expensive quantitative measurements.